Di erentiate f(x) = p 1 x+3 by using directly the de nition f0(x)  lim
x!0 f(x+
x)

Question

Dierentiate f(x) = 1/ (sqrt x+3)
by using directly the denition f'(x)
lim x→0 f(x+x)-f(x)/x

Answer 1

f(x)=1/√(x+3), f(x+Δ)=1/√(x+Δ+3).

f(x)=√(x+3)/(x+3), f(x+Δ)=√(x+Δ+3)/(x+Δ+3), when rationalised.

f(x+Δ)=√[(x+3)(1+Δ/(x+3)]/[(x+3)(1+Δ/(x+3)].

Consider (1+Δ/(x+3)^½ and 1/(1+Δ/(x+3))^-1. They approximate to:

1+½Δ/(x+3) and 1-Δ/(x+3) respectively, when Δ is very small. Therefore:

f(x+Δ)=(1+½Δ/(x+3))(1-Δ/(x+3))√(x+3)/(x+3)=(1-Δ/(x+3)+½Δ/(x+3))√(x+3)/(x+3),

f(x+Δ)=(1-½Δ/(x+3))√(x+3)/(x+3).

f(x+Δ)-f(x)=-½Δ/(x+3))√(x+3)/(x+3)=-½Δ√(x+3)/(x+3)²

Δf(x)/Δx=(f(x+Δ)-f(x))/(x+Δ-x)=-½√(x+3)/(x+3)²=-½(x+3)^½-2=-½(x+3)^-3/2 or -½/(x+3)^3/2.

So the limit as Δ→0 is -½/(x+3)^3/2=df/dx, the derivative.